We present a brief description of the “consistent discretization” approach to classical and quantum general relativity. We exhibit a classical simple example to illustrate the approach and summarize current classical and quantum applications. We also discuss the implications for the construction of a well defined quantum theory and in particular how to construct a quantum continuum limit. 1.1 C...

We quantize the generators of the little subgroup of the asymptotic Poincaré group of Lorentzian four-dimensional canonical quantum gravity in the continuum. In particular, the resulting ADM energy operator is densely defined on an appropriate Hilbert space, symmetric and essentially self-adjoint. Moreover, we prove a quantum analogue of the classical positivity of energy theorem due to Schoen ...

The mathematical structure of the continuum, in the guise of the domain of continuous, differentiable functions, has proved immensely useful in the study of nature. However, we have learned to be sceptical of any claim to the effect that our current favourite mathematical theory necessarily describes the actual structure of the physical universe. The continuous manifold of space-time may be no ...

Holographic methods are utilized to examine deviations from classical elasticity in a cellular solid, polymethacrylamide closed cell foam. A square cross section bar is subjected to static torsional deformation. The warp deformation is observed to be less in a foam bar than in a homogeneous polymeric bar used as a control. The homogeneous bar obeys the predictions of classical elasticity. Behav...

A discussion concerning the opposition between discretness and continuum in quantum mechanics is presented. In particular this duality is shown to be present not only in the early days of the theory, but remains actual, featuring different aspects of discretization. In particular discreteness of quantum mechanics is a key-stone for quantum information and quantum computation. A conclusion invol...